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a(n) = ((n-1)^2 + 2)*2^(n-2).
2

%I #26 Oct 07 2024 03:23:24

%S 1,3,12,44,144,432,1216,3264,8448,21248,52224,125952,299008,700416,

%T 1622016,3719168,8454144,19070976,42729472,95158272,210763776,

%U 464519168,1019215872,2227175424,4848615424,10519314432,22749904896,49056579584,105495134208,226291089408

%N a(n) = ((n-1)^2 + 2)*2^(n-2).

%C a(n) is the number of fixed polyiamonds of minimal area 2*n-1 that touch each side of a triangle formed in the triangular lattice. n designates the number of triangles that touch each side of the larger triangle.

%H Paolo Xausa, <a href="/A356888/b356888.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).

%F G.f.: -x*(6*x^2-3*x+1)/(2*x-1)^3.

%F E.g.f.: (exp(2*x)*(3 - 2*x + 4*x^2) - 3)/4. - _Stefano Spezia_, Sep 02 2022

%e a(3) = 12. Up to rotations and reflections there are 3 possibilities.

%e * * *

%e / \ / \ / \

%e / \ / \ / \

%e *-----* *-----* *-----*

%e / \ / \ / \ /#\ /#\ /#\

%e / \ / \ / \ /###\ /###\ /###\

%e *-----*-----* *-----*-----* *-----*-----*

%e /#\###/#\###/#\ /#\###/#\###/ \ / \###/#\###/ \

%e /###\#/###\#/###\ /###\#/###\#/ \ / \#/###\#/ \

%e *-----*-----*-----* *-----*-----*-----* *-----*-----*-----*

%t A356888[n_] := ((n-1)^2 + 2)*2^(n-2); Array[A356888, 30] (* or *)

%t LinearRecurrence[{6, -12, 8}, {1, 3, 12}, 30] (* _Paolo Xausa_, Oct 07 2024 *)

%Y Cf. A334551.

%K nonn,easy

%O 1,2

%A _Jack Hanke_, Sep 02 2022